The axisymmetric form, which will interest us here, is a tube of length \(L\) and radius \(R_0\). It is oriented following the axis \(z\) and \(r\) represent the radial axis. The reduced domain, named \(\Omega_s^*\) is represented by the dotted line. So the domain, where radial displacement \(\eta_s\) is calculated, is \(\Omega_s^*=\lbrack0,L\rbrack\).

We introduce then \(\Omega_s^{'*}\), where we also need to estimate a radial displacement as before. The unique variance is this displacement direction.

Figure 1 : Geometry of the reduce model

The mathematical problem associated to this reduce model can be describe as

where \(\eta_s\), the radial displacement that satisfy this equation, \(k\) is the Timoshenko’s correction factor, and \(\gamma_v\) is a viscoelasticity parameter. The material is defined by its density \(\rho_s^*\), its Young’s modulus \(E_s\), its Poisson’s ratio \(\nu_s\) and its shear modulus \(G_s\)

At the end, we take \( \eta_s=0\text{ on }\partial\Omega_s^*\) as a boundary condition, which will fixed the wall to its extremities.

The Maximum and minimum can be evaluated and save on .csv file. User need to define (i) <Type> ("Maximum" or "Minimum"),
(ii) "<tag>" representing this data in the .csv file, (iii) "<marker>" representing the name of marked entities and (iv) the field where extremum is computed.